By Jean-Pierre Serre
These notes are a checklist of a path given in Algiers from lOth to twenty first might, 1965. Their contents are as follows. the 1st chapters are a precis, with out proofs, of the overall homes of nilpotent, solvable, and semisimple Lie algebras. those are famous effects, for which the reader can check with, for instance, bankruptcy I of Bourbaki or my Harvard notes. the idea of complicated semisimple algebras occupies Chapters III and IV. The proofs of the most theorems are primarily whole; although, i've got additionally stumbled on it worthy to say a few complementary effects with no facts. those are indicated by way of an asterisk, and the proofs are available in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. a last bankruptcy indicates, with no evidence, the way to go from Lie algebras to Lie teams (complex-and additionally compact). it is only an creation, aimed toward guiding the reader in the direction of the topology of Lie teams and the idea of algebraic teams. i'm chuffed to thank MM. Pierre Gigord and Daniel Lehmann, who wrote up a primary draft of those notes, and likewise Mlle. Franr,:oise Pecha who used to be accountable for the typing of the manuscript.
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Additional info for Complex Semisimple Lie Algebras
However,sincesilisisomorphic to sl 2 , t 11 sl 2 = Oandhenceafortiorit 11 gil= 0. We therefore have t c ~-By writing down the condition that tis invariant under ad(Xi), we see that t is orthogonal to each ai; hence t = 0, and g is semisimple. (I) ~ is a Cartan subalgebra of g, and R is the corresponding root system. For ~ is equal to its own normalizer. The fact that R is the corresponding root system is clear. The proof of the theorem is thus complete, and with it that of Theorem 9 (the existence theorem).
And the elements i(X~~. ). One easily checks that f is a real Lie subalgebra of g, and that the Killing form off is negative. Moreover, g can be identified with the complexification f ® C of f. One says that f is a compact form of g. " When g = sl 2 , we have f = su 2 (cf. Sees. 7). Appendix. Construction of Semisimple Lie Algebras by Generators and Relations Let R be a root system in a complex vector space V. For consistency with the notation of the preceding sections, the dual of V will be denoted by ~.
One can show that, for each n;;:: 1 there is (up to isomorphism) exactly one nonreduced irreducible root system: it is the system BC" obtained as the union of the systems Bn and C" constructed above. 17. Complex Root Systems Let V be a finite-dimensional complex vector space. The definition of a symmetry given in Sec. 1 can be used without change, and Lemma 1 is still true. Hence we have the concept of a root system: Definition 7. A subset R of Vis called a (complex) root system if: (1) R is finite, spans V (as a complex vector space), and does not contain 0..